Capturing the polynomial hierarchy by second-order revised Krom logic

We study the expressive power and complexity of second-order revised Krom logic (SO-KROM$^{r}$). On ordered finite structures, we show that its existential fragment $\Sigma^1_1$-KROM$^r$ equals $\Sigma^1_1$-KROM, captures NL. all for $k\geq 1$, $\Sigma^1_{k}$ $\Sigma^1_{k+1}$-KROM$^r$ if $k$ is even, $\Pi^1_{k}$ $\Pi^1_{k+1}$-KROM$^r$ odd. The result gives an alternative to capture polynomial hierarchy. also introduce extended version (SO-EKROM). prove SO-EKROM collapses $\Pi^{1}_{2}$-EKROM $\Pi^1_1$. Both co-NP on structures.